Where to start for a Topics Class

Question

I am teaching my first non-examinable graduate course this Winter term. This course is for students ranging from Masters' students to PhD students in Pure Mathematics and is a course for students to see a topic that is a bit more specialised than the typical qualifier type course.

Since it is a bit more specialised, I am having a bit of an issue figuring out where to start. I can start with a bit of background at the risk of boring some of the more advanced PhD students working in my field, or I can tell the students that are not as adept where to find that material to catch up. I am leaning more towards the latter so that I can have more students taking the course, but the course is only 16 one-hour lectures and I want to get to some interesting material that can help someone starting to work in this field. I am also concerned I might be being overzealous in what I can accomplish in this time and might speed right through details that would be useful. (My imposter syndrome is acting up and making me feel like a lot of this stuff is trivial when maybe it's not!)

The question is this: How do you in a topics course balance giving general theory and also the more specific information that is relevant to the current research trend? I'm afraid if I spend too much time on general theory, I may end up not touching on what research is being done today. Would this be bad just spending the last lecture or two in a topics class just punting the question "What's happening nowadays in this field?"

Thanks in advance!

Answer 1

One concept/approach, which is what I try to keep in mind in addressing a heterogenous audience in a not-required more-advanced course, is exactly to not try to be "systematic", much less "complete". As noted in the question, it would be easy to use all the available time for "general theory", without ever getting around to the points of interest that motivated the course in the first place. So, instead of "laying a systematic foundation...", I try to get to interesting phenomena as directly as possible, and then back-fill "general theory" as relevant. For that matter, "general theory" is more widely available in textbooks and on-line notes than are interesting examples (since the latter tend to be written in "research papers" aiming at more-expert audiences), so students can more easily do further back-fill themselves if desired.

For that matter, some "general theory" is somehow over-hyped as a big deal, when, in fact, the introductory parts of it are really very elementary. For example, in an introduction to modular/automorphic forms (my interest...), one could position the thing as having prerequisites of "Lie theory", "algebraic number theory", "representation theory", and/or "algebraic geometry". It is certainly true that all these things are relevant and useful, but (I claim) it is not the case that one must have "systematically" studied all these things prior... For example, "Lie theory" at the entry level can be simply a study of behavior of two-by-two real and complex matrices. "Algebraic number theory" at a basic level is just about integers, or perhaps Gaussian integers, and so on. "Representation theory" (infinite-dimensional unitaries, etc.) relevant to automorphic forms will not even be found in "basic" representation theory books (which tend to do finite-dimensional and more combinatorial things), but can be approached in a down-to-earth way (for real rank-one groups) as Bargmann and Wigner did, by looking at asymptotics of solutions of second-order ordinary differential equations. (Indeed, the later work of HarishChandra and Casselman-Milicic might be construed as adapting those classical asymptotic techniques via Deligne's PDE version of the old ODE asymptotic results, as in the appendix to the C-M paper about it.)

In particular, to my perception, far too often students think that they have some moral-professional obligation to be "completely expert-prepared" in the alleged background material before starting the next stage... which is (to my perception) substantially misguided, since that way one has no inkling of what the use of the "background" might be. Trying to short-circuit this impulse a bit by directly drawing attention and explicating the applications is a good anti-dote.

And, to repeat, one has no obligation to "prove everything in class"... That's a recipe for not getting anywhere.

Answer 2

You do need to balance, and this balance will depend on several factors, primarily (i) your goals for the course, (ii) student needs, (iii) student backgrounds, and (iv) subject material. Of course (i) should take into account (ii) and (iii), and you can get a better sense of (ii) and (iii) by talking to other faculty and students beforehand.

Some advanced courses go through and prove everything or almost everything in detail assuming certain prerequisites and just get as far as they can, whereas others are more "seminar style" and just talk about ideas with little details. Most are probably somewhere in between. All these types of course can be useful but they meet different needs of students.

My main suggestion is to figure out, as concretely as possible, (i) and (ii). Then you can try to prioritize what are the most important versus least important things for the course, and how you can get there given (iii), i.e., how much theory/detail you need to/can skip.

One thing I try to do is set formal, concrete prerequisites (which I try to make as minimal as possible, but if someone wants to take the course without the prereqs, I tell them to read so-and-so before the course starts). Then I type up "appendices" or "surveys" to summarize additional preliminary material you need in the course. (Typically I won't type up proofs, but will include some examples and try to give them intuition, and include references for complete details. Usually I will lecture on most of this material as well. Depending on the course, this might come at the beginning, middle, end, or be spread out in various supplementary modules.) This way, the students who haven't learned this material already can still logically follow the course, taking these additional materials for granted.

(If you're stupid like me, you can also try to type up notes for the whole course and make them more complete than your lectures, but this take a lot of time.)